The examples presented in the previous Section REF  involved data lying near a one-dimensional underlying manifold.
Essentially the same problems can occur for data lying near higher-dimensional manifolds, and an especially illustrative and practically useful application where this situation is routinely encountered is manifold learning.
In general, manifold learning seeks to find a small collection of nonlinear coordinates that fully describe the structure of a dataset, i.e., that embed it in a lower-dimensional space.
Many techniques including kernel PCA [1]}, Laplacian eigenmaps [2]}, diffusion maps [3]}, and Isomap [4]} accomplish this via eigen-decomposition of various symmetric matrices
\(\mathbf {G} = \Phi \Lambda ^2\Phi ^T, \qquad \Phi  = \begin{bmatrix} \phi _1 & \cdots & \phi _r \end{bmatrix}\) 
